October  2012, 17(7): 2299-2311. doi: 10.3934/dcdsb.2012.17.2299

A Neumann Boundary Value Problem in Two-Ion Electro-Diffusion with Unequal Valencies

1. 

Departamento de Matemática, Universidad de Buenos Aires and CONICET, Ciudad Universitaria, Pabellón I, (1428) Buenos Aires

2. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong

3. 

Australian Research Council Centre & Excellence for Mathematics & Statistics of Complex Systems, School of Mathematics and Statistics, The University of New South Wales, Sydney, Australia

Received  December 2011 Revised  April 2012 Published  July 2012

In prior work, a series of two-point boundary value problems have been investigated for a steady state two-ion electro-diffusion model system in which the sum of the valencies $\nu_+$ and $\nu_-$ is zero. In that case, reduction is obtained to the canonical Painlevé II equation for the scaled electric field. Here, a physically important Neumann boundary value problem in the generic case when $\nu_+ + \nu_-\neq 0$ is investigated. The problem is novel in that the model equation for the electric field involves yet to be determined boundary values of the solution. A reduction of the Neumann boundary value problem in terms of elliptic functions is obtained for privileged valency ratios. A topological index argument is used to establish the existence of a solution in the general case, under the assumption $\nu_+ + \nu_- \leq 0$.
Citation: Pablo Amster, Man Kam Kwong, Colin Rogers. A Neumann Boundary Value Problem in Two-Ion Electro-Diffusion with Unequal Valencies. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2299-2311. doi: 10.3934/dcdsb.2012.17.2299
References:
[1]

W. Nernst, Zur Kinetik der in Lösung befindlichen Körper: I Theorie der Diffusion,, Z. Phys. Chem., 2 (1882), 613. Google Scholar

[2]

M. Planck, Über die Erregung von Elektricität und Wärme in Electrolyten,, Ann. Phys. Chem., 39 (1890), 161. Google Scholar

[3]

K. S. Cole, "Membranes, Ions and Impulses,", University of California Press, (1968). Google Scholar

[4]

T. L. Schwarz, "Biophysics and Physiology of Excitable Membranes", (ed. W. J. Adelman, (1971). Google Scholar

[5]

J. O'M Bokris and A. K. N. Reddy, "Modern Electrochemistry,", Plenum, (1971). Google Scholar

[6]

H. R. Leuchtag, A family of differential equations arising from multi-ion electrodiffusion,, J. Mathematical Phys., 22 (1981), 1317. Google Scholar

[7]

R. Conte, C. Rogers and W. K. Schief, Painlevé structure of a multi-ion electrodiffusion system,, J. Phys. A, 40 (2007). doi: 10.1088/1751-8113/40/48/F01. Google Scholar

[8]

H. B. Thompson, Existence for two-point boundary value problems in two-ion electrodiffusion,, J. Math. Anal. Appl, 184 (1994), 82. doi: 10.1006/jmaa.1994.1185. Google Scholar

[9]

B. M. Grafov and A. A. Chernenko, Theory of the passage of a constant current through a solution of a binary electrolyte,, Dokl. Akad. Nauk. SSR, 146 (1962), 135. Google Scholar

[10]

L. Bass, Electrical structures of interfaces in steady electrolysis,, Trans. Faraday Soc., 60 (1964), 1655. doi: 10.1039/tf9646001656. Google Scholar

[11]

N. A. Kudryashov, The second Painlevé equation as a model for the electric field in a semiconductor,, Phys. Lett. A, 233 (1997), 397. doi: 10.1016/S0375-9601(97)00545-8. Google Scholar

[12]

C. Rogers, A. Bassom and W. K. Schief, On a Painlevé II model in steady electrolysis: Application of a Bäcklund transformation,, J. Math. Anal. Appl., 240 (1999), 367. doi: 10.1006/jmaa.1999.6589. Google Scholar

[13]

L. Bass, J. Nimmo, C. Rogers and W. K. Schief, Enhanced structures of interfaces: A Painlevé II model,, Proc. Roy. Soc. London Ser. A Math. Phys. Eng. Sci., 466 (2010), 2117. doi: 10.1098/rspa.2009.0620. Google Scholar

[14]

L. Bass, Irreversible interactions between metals and electrolytes,, Proc. Roy. Soc. London A, 277 (1964), 125. doi: 10.1098/rspa.1964.0009. Google Scholar

[15]

P. Amster, M. K. Kwong and C. Rogers, On a Neumann boundary value problem for Painlevé II in two-ion electro-diffusion,, Nonlinear Analysis, (). Google Scholar

[16]

C. De Coster and P. Habets, Upper and lower solutions in the theory of ODE boundary value problems: Classical and recent results,, in, 371 (1996), 1. Google Scholar

show all references

References:
[1]

W. Nernst, Zur Kinetik der in Lösung befindlichen Körper: I Theorie der Diffusion,, Z. Phys. Chem., 2 (1882), 613. Google Scholar

[2]

M. Planck, Über die Erregung von Elektricität und Wärme in Electrolyten,, Ann. Phys. Chem., 39 (1890), 161. Google Scholar

[3]

K. S. Cole, "Membranes, Ions and Impulses,", University of California Press, (1968). Google Scholar

[4]

T. L. Schwarz, "Biophysics and Physiology of Excitable Membranes", (ed. W. J. Adelman, (1971). Google Scholar

[5]

J. O'M Bokris and A. K. N. Reddy, "Modern Electrochemistry,", Plenum, (1971). Google Scholar

[6]

H. R. Leuchtag, A family of differential equations arising from multi-ion electrodiffusion,, J. Mathematical Phys., 22 (1981), 1317. Google Scholar

[7]

R. Conte, C. Rogers and W. K. Schief, Painlevé structure of a multi-ion electrodiffusion system,, J. Phys. A, 40 (2007). doi: 10.1088/1751-8113/40/48/F01. Google Scholar

[8]

H. B. Thompson, Existence for two-point boundary value problems in two-ion electrodiffusion,, J. Math. Anal. Appl, 184 (1994), 82. doi: 10.1006/jmaa.1994.1185. Google Scholar

[9]

B. M. Grafov and A. A. Chernenko, Theory of the passage of a constant current through a solution of a binary electrolyte,, Dokl. Akad. Nauk. SSR, 146 (1962), 135. Google Scholar

[10]

L. Bass, Electrical structures of interfaces in steady electrolysis,, Trans. Faraday Soc., 60 (1964), 1655. doi: 10.1039/tf9646001656. Google Scholar

[11]

N. A. Kudryashov, The second Painlevé equation as a model for the electric field in a semiconductor,, Phys. Lett. A, 233 (1997), 397. doi: 10.1016/S0375-9601(97)00545-8. Google Scholar

[12]

C. Rogers, A. Bassom and W. K. Schief, On a Painlevé II model in steady electrolysis: Application of a Bäcklund transformation,, J. Math. Anal. Appl., 240 (1999), 367. doi: 10.1006/jmaa.1999.6589. Google Scholar

[13]

L. Bass, J. Nimmo, C. Rogers and W. K. Schief, Enhanced structures of interfaces: A Painlevé II model,, Proc. Roy. Soc. London Ser. A Math. Phys. Eng. Sci., 466 (2010), 2117. doi: 10.1098/rspa.2009.0620. Google Scholar

[14]

L. Bass, Irreversible interactions between metals and electrolytes,, Proc. Roy. Soc. London A, 277 (1964), 125. doi: 10.1098/rspa.1964.0009. Google Scholar

[15]

P. Amster, M. K. Kwong and C. Rogers, On a Neumann boundary value problem for Painlevé II in two-ion electro-diffusion,, Nonlinear Analysis, (). Google Scholar

[16]

C. De Coster and P. Habets, Upper and lower solutions in the theory of ODE boundary value problems: Classical and recent results,, in, 371 (1996), 1. Google Scholar

[1]

Eugenio Montefusco, Benedetta Pellacci, Gianmaria Verzini. Fractional diffusion with Neumann boundary conditions: The logistic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2175-2202. doi: 10.3934/dcdsb.2013.18.2175

[2]

Annegret Glitzky. Energy estimates for electro-reaction-diffusion systems with partly fast kinetics. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 159-174. doi: 10.3934/dcds.2009.25.159

[3]

Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935

[4]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

[5]

Elisa Sovrano. Ambrosetti-Prodi type result to a Neumann problem via a topological approach. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 345-355. doi: 10.3934/dcdss.2018019

[6]

Pablo Amster, Colin Rogers. On a Ermakov-Painlevé II reduction in three-ion electrodiffusion. A Dirichlet boundary value problem. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3277-3292. doi: 10.3934/dcds.2015.35.3277

[7]

Sergiu Aizicovici, Nikolaos S. Papageorgiou, V. Staicu. The spectrum and an index formula for the Neumann $p-$Laplacian and multiple solutions for problems with a crossing nonlinearity. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 431-456. doi: 10.3934/dcds.2009.25.431

[8]

Kin Ming Hui, Sunghoon Kim. Existence of Neumann and singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4859-4887. doi: 10.3934/dcds.2015.35.4859

[9]

T. J. Christiansen. Resonances and balls in obstacle scattering with Neumann boundary conditions. Inverse Problems & Imaging, 2008, 2 (3) : 335-340. doi: 10.3934/ipi.2008.2.335

[10]

Rúben Sousa, Semyon Yakubovich. The spectral expansion approach to index transforms and connections with the theory of diffusion processes. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2351-2378. doi: 10.3934/cpaa.2018112

[11]

Haitao Yang, Yibin Zhang. Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5467-5502. doi: 10.3934/dcds.2017238

[12]

Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763

[13]

Zhenhua Zhang. Asymptotic behavior of solutions to the phase-field equations with neumann boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 683-693. doi: 10.3934/cpaa.2005.4.683

[14]

Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018

[15]

Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez. Parabolic problems with varying operators and Dirichlet and Neumann boundary conditions on varying sets. Conference Publications, 2007, 2007 (Special) : 181-190. doi: 10.3934/proc.2007.2007.181

[16]

Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190

[17]

Zhongwei Tang. Segregated peak solutions of coupled Schrödinger systems with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5299-5323. doi: 10.3934/dcds.2014.34.5299

[18]

Shengbing Deng, Fethi Mahmoudi, Monica Musso. Bubbling on boundary submanifolds for a semilinear Neumann problem near high critical exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3035-3076. doi: 10.3934/dcds.2016.36.3035

[19]

Jason Metcalfe, Jacob Perry. Global solutions to quasilinear wave equations in homogeneous waveguides with Neumann boundary conditions. Communications on Pure & Applied Analysis, 2012, 11 (2) : 547-556. doi: 10.3934/cpaa.2012.11.547

[20]

Julii A. Dubinskii. Complex Neumann type boundary problem and decomposition of Lebesgue spaces. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 201-210. doi: 10.3934/dcds.2004.10.201

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]